Trigonometry Examples

{log}_{81} (9)

$\log_{81} (9)$

Step 1

Rewrite as an equation.

{log}_{81} (9) = x

$\log_{81} (9) = x$

Step 2

Rewrite

{log}_{81} (9) = x

$\log_{81} (9) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$ does not equal

1

$1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

81^{x} = 9

$81^{x} = 9$

Step 3

Create expressions in the equation that all have equal bases.

{(3^{4})}^{x} = 3^{2}

${(3^{4})}^{x} = 3^{2}$

Step 4

Rewrite

{(3^{4})}^{x}

${(3^{4})}^{x}$ as

3^{4 x}

$3^{4 x}$ .

3^{4 x} = 3^{2}

$3^{4 x} = 3^{2}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

4 x = 2

$4 x = 2$

Step 6

Solve for

x

$x$ .

x = \frac{1}{2}

$x = \frac{1}{2}$

Step 7

The variable

x

$x$ is equal to

\frac{1}{2}

$\frac{1}{2}$ .

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Step 7.1

Factor

2

$2$ out of

2

$2$ .

\frac{2 (1)}{4}

$\frac{2 (1)}{4}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 \cdot 1}{2 \cdot 2}

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$